A direct proof of the equivalence between Dirichlet's principle and Perron's method

2025/11/24 by Gantumur, Tsogtgerel
#35J05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.18762

Abstract

We give a short proof that for a bounded domain Ω⊂ℝⁿ and continuous boundary data g∈ C(∂Ω) admitting a continuous finite-energy extension ϕ∈ H¹(Ω)∩ C(Ω), the minimizer of the Dirichlet energy E(v) = ∫_Ω |∇ v|² dx, v-ϕ∈ H¹₀(Ω), coincides with the Perron solution h_g of the Dirichlet problem Δu = 0 in Ω with boundary data g. The argument stays entirely in H¹(Ω) and uses only strong convergence via strict convexity of the Dirichlet energy, Friedrichs' inequality, Weyl's lemma, and Wiener's exhaustion by regular subdomains. No weak convergence, Poisson problems with distributional right hand sides, or general elliptic theory are needed.

Citations

• The Perron solution for elliptic equations without the maximum principle

Related

• The Perron solution for elliptic equations without the maximum principle
• The Perron method associated with finely p-harmonic functions on finely open sets
• On the Perron solution of the caloric Dirichlet problem: an elementary approach
• A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate-elliptic operators via holomorphic maps
• Variational solutions of the Dirichlet problem, Lebesgue's cusp and non-local properties